they are founded on fanciful, or even false opinions. It may be true, that supines are mere nouns; that deponents are really passive verbs; and the like: yet no disadvantage arises from the use of such terms as supines and deponents to mark those cases. Or rather, it would be a great disadvantage that the learner should not know the meaning of such terms; so as to be able to understand those grammatical discussions in which they are employed. It may be that many of the Rules given in the common books of Arithmetic are arbitrary and superfluous, and that they might all be reduced, with advantage, to a smaller number. As they are given, they serve at least to classify and multiply examples of numerical operations, and are themselves multiplied examples of the simpler and more comprehensive principles to which the student is afterwards to be led, when his mind is matured and prepared for dealing with such principles. The boy learns rules as rules, which he can do easily and well; and the youth learns reasons and principles all the more easily, because the process of learning rules has preceded. But these considerations belong rather to the mode of teaching, which will be a subject for our attention hereafter.

113 It may be said that by thus defending and commending the use of the rules and technicalities belonging to the old methods of teaching, we do not adequately appreciate the great recent improvements in education; the new views of grammar and of the relations of language, and of the foundations and reasons of arithmetic, algebra, and geometry;—and the simplicity and clearness which have been introduced into the teaching of these subjects. To this I reply, that the new views of the fundamental principles, both of philology and of mathematics which have recently been published, have been, as I believe, very efficacious in promoting a better understanding of those subjects; but that they have not produced this effect merely in virtue

of their being better views, superseding worse, but in virtue of their being the results of the activity of thought and research of the Teachers. For though technical phrases and rules and maxims may be very useful instruments of education before they are fully understood by the learners, they cannot be used with any great efficacy for such purposes, except they are understood by the Teachers: and the new views of recent speculators, regarding language and antiquities, geometry and algebra, have been the results of their endeavours to ascertain fully the significance, truth, and foundations of the doctrines which the traditional forms of classical and mathematical literature take for granted. Precisely because the new doctrines are expressions of advances towards clear insight and full conviction in the minds of the Teachers, they are better doctrines for them, and enable them to teach better, than, without such an intellectual movement going on among them, they could have done. Such a mental advance will make their instructions both more rationally sound, and more zealous and persevering, than they would otherwise be. I believe it will be found that this is the source of any greater effectiveness of modern systems of teaching classics and mathematics which may have occurred, rather than any virtue inherent in the new methods themselves. When the expressions which convey the new views have come, in their turn, to be familiar technicalities, dimly understood; and when the new methods are applied by a number of teachers, of ordinary zeal and intelligence, to learners of all variable degrees of capacity, it is probable that the average success of the new methods will not be much greater than that of the old ones; and certainly it does not appear likely that the new methods will produce better scholars and better mathematicians, either in the most eminent cases or on the average, than the old methods produced. There appears to be no disadvantage, but rather a con

siderable means of instruction, in having our education consist of ancient methods, which though sound and good, may be simplified, extended, and clothed in a deeper significance by newer methods which the teacher himself suggests. The new method comes as a commentary upon the old; and gives to the education so conducted the combined advantages of the stability of a fixed system, and the vivacity of a present reform.

114 It is, therefore, with no want of admiration for the subtilty and comprehensiveness of intellect which has been shown in many recent views of Algebra and Geometry, that I recommend our adhering to the ancient methods of treating such subjects, so far as the general purposes of a liberal education are concerned. Such views are truly admirable, as corrections, and extensions, or it may be as the antithesis of the established and traditional modes of treating the subjects; but if they were to become themselves established and traditional, they would be (as I have already endeavoured to show) far less effective in the discipline of the reason, than the older methods; besides depriving us of the continuity of that intellectual tradition which I have already spoken of, as one of the great ends of mathematical teaching in the course of a liberal education. I recommend the rejection, in our ordinary educational system, of the many novelties in notation and expression which have recently appeared in our Cambridge Mathematical works; but I admire the mathematical talents of those who have produced these works; and I think that such speculations are both very remarkable manifestations of mathematical skill and thought, and very fit subjects of attention for our mathematical students, where they reach the higher stages of their progress.

115 There is one leading question, in such an education as we are contemplating, on which I have already spoken, but on which it may not be useless to

add a few words:-I mean, the question whether both mathematical and classical instruction should be considered necessary in the case of every student. It is sometimes said that we shall educate men better, by encouraging in each that study for which he has talent and inclination;-not tormenting the man of classical taste with fruitless lessons of algebra, or the man of mathematical intellect with obscure passages of Greek. It is said, sometimes, that by such a genial education alone, do we really educate the man, or bring out his genius;-that the seeming of mathematical prowess, or of classical learning, which we wring by force from ungenial and unwilling minds, is of no value, and is no real culture. But to this we reply, that if men come really to understand Greek or Geometry, there is then, in each study, a real intellectual culture, however unwillingly it may have been entered upon. There can be no culture without some labour and effort; to some persons, all labour and effort are unwelcome; and such persons cannot be educated at all, without putting some constraint upon their inclinations. No education can be considered as liberal, which does not cultivate both the Faculty of Reason and the Faculty of Language; one of which is cultivated by the study of mathematics, and the other by the study of classics. To allow the student to omit one of these, is to leave him half educated. If a person cannot receive such culture, he remains, in the one case, irrational, in the other illiterate, and cannot be held up as a liberally educated person. To allow a person to follow one of these lines of study, to the entire neglect of the other, is not to educate him. It may draw out his special personal propensities; but it does not draw out his general human Faculties of Reason and Language. The object of a liberal education is, not to make men eminently learned or profound in some one department, but to educe all the faculties by which man shares in

the highest thoughts and feelings of his species. It is to make men truly men, rather than to make them men of genius, which no education can make them.

116 But even with regard to men of genius, it is not true that they have generally been men of one kind of cultivation only, or capable only of one kind of intellectual excellence. The case has been quite the reverse. During the middle ages, and down to the last century, the greatest mathematicians were almost invariably good classical scholars; and good scholars were almost invariably well acquainted with mathematical literature, and often very fond of it. And this connexion, in the main, has continued to our own day, so far as the mathematics and classics belonging to a liberal education are concerned. Not to speak of living persons, whose career at Cambridge might be adduced to prove this, the greatest Greek scholar of the last generation, Porson, was fond of Algebra, and was a proficient in it;—and if we run over the highest wranglers of the last sixty years, we find at every period, men known to be well versed in classical literature, as Otter, Brinkley, Outram, Raincock, Wrangham, Palmer, T. Jackson, R. Grant, and many others.

117 Indeed, there can be no doubt but that the clearness of mind and vigour of character which make a man eminent in one line of study will also enable him to master the elementary difficulties of another subject, if it is fairly brought before him as something which must be done; although, if it be presented to him as a matter of choice whether he will make the attempt, caprice, fastidiousness, and the pleasure of doing what he can already do easily and well, may make him turn with repugnance from a subject in which he has not learned to feel any interest.

118 To which we may add, that to be able to command the attention and direct the mental powers, so as to master a subject which is not particularly